Data with impact and DIF | The covariate belongs in the ability model | DIF: one test per item | Purification | A covariate the fit does not have | Covariates of ability | Covariates that vary within person

Purpose | 4PL MIRT model | Observation model | $$\Pr(Y_{ij} = 1 \mid \theta_i, \psi_j) | $$\ell_{ij}(\theta_i,\psi_j) | Parameterization used in bigIRT | Latent distribution | Marginal likelihood target | Exact marginal likelihood | $$\log L(\psi) | Why Laplace | Per-person posterior approximation | $$f_i(\theta_i;\psi) | $$H_i(\psi) | Direct Laplace objective used in bigIRT | $$\tilde L(\psi) | Value versus gradient | Direct value | Approximate gradient | Item-side frozen-posterior gradient | Why this can be reasonable | Ability mean gradient | Ability correlation gradient | Why optimize in one direct stage? | Advantages | Disadvantages | Current algorithm in words | Identification | Why the approximation can still work well | 1. The value tracks the intended target | 2. The frozen-posterior gradient is locally sensible | 3. Warm starts help | 4. Large-data settings help the local Gaussian approximation | Important limitations | Approximate direct gradient | Sensitivity of $c$ and $d$ | Dependence on latent-distribution specification | Expensive evaluations | Alternative approaches | 1. Two-step Laplace EM | Pros | Cons | 2. Exact direct Laplace gradient via implicit differentiation | 3. Adaptive quadrature | 4. MHRM and related stochastic marginal methods | 5. Variational approximations | 6. JML / conditional optimization | When laplace_direct is attractive | When another method may be preferable | Summary | Measured behaviour of the frozen-posterior gradient | What the weight is actually worth | Why the near-correct weight loses, and what cannot fix it | It is not the quality of the Gaussian approximation | It is a bias-variance trade, concentrated where identification is weak | Scaling the log-determinant cannot substitute for it | What follows | Resolution: the gradient was right, the objective was incomplete | What this retracts